Manipulation of the dynamics of many-body systems via quantum control methods

We investigate how dynamical decoupling methods may be used to manipulate the time evolution of quantum many-body systems. These methods consist of sequences of external control operations designed to induce a desired dynamics. The systems considered for the analysis are one-dimensional spin-1/2 models, which, according to the parameters of the Hamiltonian, may be in the integrable or non-integrable limits, and in the gapped or gapless phases. We show that an appropriate control sequence may lead a chaotic chain to evolve as an integrable chain and a system in the gapless phase to behave as a system in the gapped phase. A key ingredient for the control schemes developed here is the possibility to use, in the same sequence, different time intervals between control operations.


I. INTRODUCTION
The study of one-dimensional quantum many-body systems out of equilibrium offers several fundamental challenges. A main question is whether an isolated quantum system can or not thermalize after a quench. While a relationship between quantum chaos and thermalization is certain to exist [1][2][3][4][5][6], an ongoing controversy is the occurrence of thermalization for a chaotic system in the gapped phase [7][8][9]. In terms of transport properties, the discussions are associated with the expected dynamical behavior of systems in the integrable and nonintegrable regimes [10][11][12][13][14][15][16][17][18][19][20], and in the gapped and gapless phases [21,22]. Extraordinary technological advances have motivated the use of experiments to address these issues. In this work we show how these experiments may benefit from dynamical decoupling methods to accurately reach the different regimes and phases of interest.
The fundamental questions raised above have found a test-bed in experiments with ultracold quantum gases in optical lattices [23] and with magnetic compounds [24][25][26]. Both systems require a high level of controllability. In optical lattices, control of atomic transport is achieved with time-dependent variations of the depth and phase of the lattice potential; whereas in magnetic systems the preparation of samples where impurities and unwanted couplings are negligible is key for obtaining a desired transport behavior. In addition to monitoring experimental conditions, we advocate the use of sequences of external electromagnetic field pulses to manipulate the dynamics of the systems. This method of quantum control has long been employed in nuclear magnetic resonance (NMR) spectroscopy [27,28]. There, the Hamiltonian of nuclear spin systems is modified with sequences of unitary transformations (radio-frequency pulses), which constantly rotate the spins and remove or rescale selected terms of the original Hamiltonian. These techniques have acquired a new dimension in the context of quantum information processing [29,30], where the studies of transport of information [31] and the manipulation of electron spins with ultrafast laser pulses [32,33] are among the latest developments. Recently, these methods have been proposed also for the control of atomic transport in spinor optical lattices [34].
In the present work we use the theory developed in NMR to construct sequences of control operations that appropriately modify the Hamiltonian of a one-dimensional quantum many-body system and forces it to follow a desired dynamics. The systems analyzed are modeled by spin-1/2 chains, which may also be mapped onto systems of spinless fermions or hardcore bosons. We focus on limits that are of interest for studies of transport behavior and thermalization. Specifically, our goals are (i) to dynamically remove the effects of terms leading to the onset of chaos, such as disorder and frustration, and compel the chain to evolve as an integrable system, as introduced in Refs. [18,19]; (ii) to dynamically increase the effects of terms that can drive the system across a transition from the gapless to the gapped phase (and vice-versa), aiming, for example, at forcing a conductor to behave as an insulator or at bringing a chaotic system into the gapped phase. For the gapless-gapped transition, the strengths of the coupling terms in the Hamiltonian are modified by applying to the target system sequences with different time intervals between the control operations. The idea of varying the time separation between pulses has been exploited in sequences created to suppress the couplings between a qubit and its environment [35,36]. Contrary to that, the systems considered here are isolated from the environment and our objective is not necessarily to completely eliminate couplings, but instead to control their strengths. This paper is organized as follows. Sec. II describes the model under investigation. Sec. III reviews a technique to design control sequences, which is based on the average Hamiltonian theory. Sec. IV presents in detail sequences that dynamically lead the system studied across a transition between the nonintegrable and integrable limits and between the gapless and gapped phases. Numerical results are shown for the time evolution of local magnetization based on a particular initial state. A discussion about control metrics and the dependence of the outcomes on initial states is left for the Appendix. Conclusions are given in Sec. V.

II. DESCRIPTION OF THE MODEL
We study a one-dimensional spin-1/2 system with open boundary conditions described by the Hamiltonian where Above, is set equal to 1, L is the number of sites, and S x,y,z n = σ x,y,z n /2 are the spin operators at site n, σ x,y,z n being the Pauli matrices.
(i) The parameter ǫ n corresponds to the Zeeman splitting of spin n, as determined by a static magnetic field in the z direction. The system is clean when all ǫ n 's are equal (ǫ n = ǫ) and it is disordered when at least one spin n has an energy splitting different from the others (ǫ n = ǫ). We therefore consider only on-site disorder.
(ii) H NN , also known as the XXZ Hamiltonian, corresponds to the nearest-neighbor (NN) exchange; S z n S z n+1 is the Ising interaction and S x n S x n+1 + S y n S y n+1 is the flip-flop term. The coupling strength J and the anisotropy ∆ are assumed positive, thus favoring antiferromagnetic order. By varying ∆ in H 0 = H N N we may go from the gapless phase (∆ < 1) to the gapped phase (∆ > 1), which, at T = 0, is followed by the metal-insulator Mott transition [37]. In the particular case of ∆ = 0, we deal with the XY model; whereas ∆ = 1 leads to the isotropic Heisenberg Hamiltonian [38].
(iii) The parameter α refers to the ratio between the next-nearest-neighbor (NNN) exchange, as determined by H NNN , and the NN couplings. The inclusion of NNN antiferromagnetic exchange frustrates the chain, since NN exchange favors ferromagnetic alignment between the second neighbors. Different phenomena are expected in the presence of frustration [39]. In the case of a closed isotropic system, for instance, a critical value α c ≈ 0.241 exists which separates the spin fluid phase (α < α c ) from the dimer phase (α > α c ) [40,41].  [42]. The addition of defects [43], even if just one in the middle of the chain [44], or the inclusion of NNN couplings [45,46] may lead to the onset of quantum chaos.

III. DESIGN OF DYNAMICAL DECOUPLING SEQUENCES
Dynamical decoupling methods consist in the application of sequences of unitary transformations to generate a desired effective propagator at time t. The first experimental implementations of sequences of electromagnetic pulses aiming at controlling the dynamics of molecular systems appeared in the context of NMR spectroscopy and correspond to the so-called spin-echo and Carr-Purcell sequences [47,48]. The basic idea of the method is to add a time-dependent control Hamiltonian, H c (t), to the original (time-independent, in our case) Hamiltonian of the system, H 0 , so that the system evolution becomes dictated by the following propagator where T stands for time ordering. This operator may also be written as [27,28], depends only on the introduced perturbation H c (t) and is known as the control propagator. We deal with cyclic control sequences with cycle time T c . This means that the perturbation and the control propagator are periodic, that Therefore, the evolution of the system at any time T n = nT c requires only the knowledge of the propagator after one cycle [27,28].
We consider the ideal scenario of arbitrarily strong and instantaneous control operations, known in NMR as hard pulses and popularized in the field of quantum information as bang-bang control [29]. Each pulse P k+1 is applied after a time interval of free evolution τ k+1 = t k+1 − t k , where k ∈ AE and t 0 = 0. For a cyclic sequence with m pulses, the evolution operator at T c = mτ is then given by, (2) Above, the notation H m−1 = (P m−1 . . . P 1 ) † H 0 (P m−1 . . . P 1 ) has been used. The HamiltonianH in the last line is obtained via Baker-Campbell-Hausdorff expansion and is referred to as the average Hamiltonian [27,28]. The lowest order term ofH in τ corresponds tō In designing a dynamical decoupling sequence, the primary goal is to guarantee the proximity ofH (0) to the desired Hamiltonian. This is the focus of the current work. We note, however, that various strategies exist to eliminate some of the higher order terms [27,28] or to reduce the accumulation of errors arising from imperfect averaging [49][50][51][52].

IV. MODIFYING THE SYSTEM EVOLUTION
In this section we explore how dynamical decoupling schemes may induce transitions between different regimes and phases of a quantum many-body system. We aim at dynamically reshaping the original Hamiltonian (1) to make it as close as possible to a desired Hamiltonian H w (the subscript "w" is used whenever we refer to parameters and quantities associated with the wanted evolution). Our numerical results are performend in the full Hilbert space of dimension D = 2 L . We cannot take advantage of the symmetries of H 0 (1), such as conservation of total spin in the z direction, to reduce this dimension, because the intervals of free evolution of the pulsed system involve Hamiltonians [H k in Eq. (3)] that are different from the original H 0 .

A. Non-integrable vs integrable systems
To start, we discuss sequences of control pulses that average out the terms in Hamiltonian (1) that may lead to the onset of chaos, namely on-site disorder and NNN exchange. The first and simplest sequence that we present was proposed in [18,19] and it aims at eliminating the effects of on-site disorder. Consider a disordered system described by H z + H N N with one or more sites having ǫ n = ǫ. In order to recover the transport behavior of a clean chain given by H N N , we apply, after every τ , control operations P , which rotate all spins by 180 o (π-pulses) around a direction perpendicular to z. Around x, for instance, the pulses are The propagator at cycle time T c = 2τ becomes which leads to the following dominant term of the average Hamiltonian as desired.

Annihilating the effects of NNN exchange: HNN + HNNN → HNN
A sequence to eliminate the effects of NNN exchange (and also on-site disorder) was introduced in [18] and is further investigated here. It has eight π-pulses equally separated (T c = 8τ ) and applied to selected spins, which therefore assumes site addressability. The pulses are They change the signs of the coupling terms in each interval of free evolution according to Table II.
The sequence removes NNN couplings, but at the price of also reducing the effects of NN couplings. We achieve the following dominant term for the average Hamiltonian, In Fig. 1, we show the evolution of a chaotic chain described by H N N + H N N N under free evolution and subjected to the pulses (5), and compare it with the dynamics of an ideal integrable system given by H N N /2. The quantity considered is the local magnetization, which is defined as the magnetization of the first half of the chain, We choose as initial state |Ψ(0) a highly excited state far from equilibrium: it consists of spins pointing up in the first half of the chain and pointing down in the other half, so that M (0) = L/4. A discussion about control metrics and the dependence of the results on different initial states is given in the Appendix. In order to average out unwanted terms of the Hamiltonian, we need to consider cycle times smaller than the time scale of the free evolution of the system, that is, JT c < 1 [53]. For the T c 's considered in Fig. 1, the agreement between the evolutions of the chaotic system under control pulses and the integrable chain is excellent. As the values of τ increase, higher order terms in the average Hamiltonian become significant, specially at long times, and the pulsed system must eventually separate from the ideal scenario. Even though strategies exist to reduce the effects of higher order terms inH [49][50][51][52], the access to short time intervals between pulses is essential to the success of dynamical decoupling methods.
Note that control sequences are designed to match the aspired evolution at the completion of cycles, but not necessarily in between cycles. This is well illustrated in the inset. The symbols indicate data obtained at every T n = nT c , while the solid lines correspond to the values of M (t) at every t n = nτ . The largest mismatches between the evolution determined by H N N + H N N N + H c (t) and by H N N /2 occur in between cycles.
A second alternative to recover the dynamical behavior of an integrable system consists in canceling out the NN exchange leaving the NNN couplings unaffected. This is achieved with a sequence of four pulses equally separated (T c = 4τ ), which alternates rotations around perpendicular directions, say x and y. The pulses can affect only spins in odd sites, or only in even sites, or they alternate between odd and even sites as below, The pulses (7) change the signs of the terms of H N N + H N N N as shown in Table III.  (7).
The above control sequence leads to two uncoupled chains, one consisting of exchange couplings between spins in the original odd sites and the other one consisting of couplings between spins in the original even sites. The first order term in the average Hamiltonian is simply,H (0) = H N N N .

B. Gapped vs gapless systems
Here, we analyze how the time evolution typical of a system with an energy gap may be achieved from a gapless system subjected to a quantum control scheme. Instead of completely averaging out unwanted terms of the Hamiltonian, as done in the previous section, the goal now is to change the ratio between different coupling strengths. This is accomplished by varying the time interval between pulses, so that the weight of some specific terms may be decreased relatively to others.
1. Increasing the role of the Ising interaction: ∆ < 1 → ∆ > 1 Consider a given system in the gapless phase described by H N N , where ∆ < 1. To induce the transport behavior of a chain in the gapped phase, we develop a sequence that leaves the Ising interaction unaffected, but reduces the strength of the flip-flop term. The sequence consists of two π-rotations around the z axis, which affect only spins in even sites (or only in odd sites), that is, The first pulse is applied after τ 1 and the second after τ 2 , which results in the following reshaped Hamiltonian at The gapped phase is obtained if T c ∆/(τ 1 − τ 2 ) > 1.
On the left panels of Fig. 2 we compare the desired evolution of a gapped system described by H N N /4, where ∆ w = 2, with the dynamics of a given gapless system described by H N N , where ∆ = 1/2, and subjected to control operations. The pulses, given by Eq. (8), have time intervals that lead to T c ∆/(τ 1 − τ 2 ) = 2. The agreement between the two cases at the completion of each cycle is very good for the chosen time intervals, but of course it worsens as T c increases. The price to dynamically achieve a transition from a gapless to a gapped Hamiltonian, however, is the slowing down of the system evolution in the gapped phase by a factor of 4. For a discussion about ballistic vs diffusive transport in XXZ models with small and large values of ∆, see Refs. [21,22] and references therein. It is also possible to force a transition from a gapped to a gapless phase. For example, pulses (7) separated by different intervals τ 1 , τ 2 = τ 4 , and τ 3 , reshape the Hamiltonian of a Mott insulator described by H N N with ∆ > 1 as follows,H The gapless phase is then obtained if ∆(τ 1 + τ 3 − 2τ 2 ) < (τ 1 − τ 3 ). In particular, when τ 1 + τ 3 = 2τ 2 we recover the XY model.
2. From the spin fluid to the dimer phase: α < αc → α > αc Let us now focus on a system described by H N N + αH N N N with α < α c and ∆ = 1. Our goal is to apply to it a sequence of control pulses that induces a dynamical behavior similar to that of a system in the dimerized phase (α > α c ). A possible sequence consists of the four pulses presented in Eq. (7), although now separated by different time intervals τ 1 and τ 2 = τ 3 = τ 4 = τ a . It leads to the following dominant term of the average Hamiltonian at The desired dynamics is reached when αT c /(τ 1 − τ a ) > α c . On the right panels of Fig. 2, we compare the dynamics of a system in the gapless phase given by H N N + αH N N N , where α = 0.1 and ∆ = 1, with the wanted dynamics of a system in the gapped phase described by (H N N + αH N N N )/6.4, where α w = 0.64 and ∆ = 1. The pulses (7) applied to the original system are separated by intervals that satisfy the constraint αT c /(τ 1 − τ a ) = 0.64. For JT c < 1, the dynamics of the perturbed gapless system at the completion of the cycles agrees well with the evolution of the gapped system.
Notice that our pulsed system will now be in the limit of strong frustration and will therefore become chaotic [46]. We expect an initial state with energy far from the edges of the spectrum to show a dynamical behavior typical of a chaotic system [54], although slowed down by a factor α w /α.
A variety of new pulse sequences may be designed according to our needs and experimental capabilities. For instance, the sequence described above, which has no effect on NNN couplings, could be combined with a sequence of π-pulses in the z direction given by These pulses would affect only the flip-flop term of H N N N . The combined sequence would reduce the strength of H N N and keep only the Ising interaction from H N N N . Hamiltonians similar to this resulting one have been analyzed in the context of transport [55] and thermalization in gapped systems [9].

V. CONCLUSION
We studied how quantum control methods may be used to manipulate the dynamics of quantum many-body systems. This is accomplished with sequences of unitary transformations designed to achieve a desired evolution operator. The unitary transformations correspond to instantaneous electromagnetic field pulses, which, in the case of spin systems, rotate the spins by a certain angle. The strategy employed to construct the control sequences was the average Hamiltonian theory.
Four target spin-1/2 chains were considered. Two were chaotic: (i) disordered with NN exchange, (ii) clean with NN exchange and NNN frustrating couplings; and two were in the gapless phase: (iii) clean with NN couplings and small Ising interaction, (iv) clean with NN couplings and weak frustration. We discussed sequences of control pulses aiming at averaging out the effects of disorder, in case (i), and the effects of NNN couplings, in case (ii), and then induce in these chaotic systems a dynamical behavior typical of an integrable clean chain with only NN exchange. We also developed new sequences capable of changing the ratio between the coupling strengths of the system. This was done by exploiting, in the same sequence, different time intervals between the pulses, so that the effects of specific terms in the Hamiltonian may be reduced, but not necessarily eliminated. By dynamically decreasing the weight of the flip-flop term in (iii), we obtained a time evolution characteristic of gapped systems with large Ising interaction (also mentioned was a scheme to achieve the dynamics of a metal out of an insulator). By decreasing the strength of the NN couplings with respect to the NNN exchange in (iv), we obtained the dynamical behavior of a chain in the limit of strong frustration (also mentioned was a way to combine to this scheme a sequence that removes the NNN flip-flop term).
Site addressability and the possibility to vary the time intervals between pulses add versatility to dynamical decoupling schemes and open an array of new opportunities for controlling quantum systems. Our focus here was to use these ideas as an additional tool to reach regimes and phases that are relevant in studies of the nonequilibrium dynamics of many-body systems.

Appendix A: Control Metric
In the main text, we showed numerical results for the local magnetization. This choice was motivated by works on transport properties and thermalization, where quantities of experimental interest are usually considered. The decision to take the particular initial state where all up-spins are in the first half of the chain was a way to connect our results with previous transport studies [17][18][19], and also to provide a good illustration, since states where M (0) ∼ 0 would not give much information. However, in order to quantify how successful the designed control sequences are in general, we need an appropriate control metric which is independent of the initial state.
How close a pulse sequence brings the evolution of a system to a desired dynamics given by U w (t) may be assessed by the so-called propagator fidelity, which is defined as [56], Above, D = 2 L is the dimension of the system and U (t) dictates the evolution of the pulsed system. This quantity deals directly with propagators and therefore avoid any mention to particular initial states. Our goal is to achieve F u (t) → 1. In Fig. 3, we show again the cases studied in Figs. 1 and 2, but now from the perspective of the evolution operator. The performance of the pulse sequences deteriorates for longer T c 's and longer times. For the shortest cycle times considered, however, the agreement between the pulsed and desired dynamics is very good for relatively long times, giving a proof of principle that indeed we may manipulate the dynamics of quantum many-body systems with dynamical decoupling methods [57].
The agreement between U w (t) and U (t) is best at the completion of cycles, as shown in the left panel and also as verified, but not shown, for the system in the right panel. Interestingly, however, for the system in the middle panel this does not always hold. In the middle panel we compare the gapless system H N N with ∆ = 1/2 and the gapped system H w = H N N /4 with ∆ = 2. The applied pulses involve a single direction z and the cycle is completed after only two pulses [cf. Eq. (8)]. In the figure, we show with circles the values of F u after every pulse, so each value of the pair τ 1,2 has associated with it two curves. For short times, the performance of the sequence is best at the completion of the cycles, as expected, but at longer times, the scenario is reversed and the best performance appears in between cycles. The origin of this shift of roles, which occurs faster for longer cycle times, deserves further investigation.